A jacobian inequality for gradient maps on the sphere and its application to directional statistics

Tomonari Sei

Research output: Contribution to journalArticle

Abstract

In the field of optimal transport theory, an optimal map is known to be a gradient map of a potential function satisfying cost-convexity. In this article, the Jacobian determinant of a gradient map is shown to be log-concave with respect to a convex combination of the potential functions when the underlying manifold is the sphere and the cost function is the distance squared. As an application to statistics, a new family of probability densities on the sphere is defined in terms of cost-convex functions. The log-concave property of the likelihood function follows from the inequality.

Original languageEnglish
Pages (from-to)2525-2542
Number of pages18
JournalCommunications in Statistics - Theory and Methods
Volume42
Issue number14
DOIs
Publication statusPublished - 2013 Jul 18

Keywords

  • Directional statistics
  • Gradient map
  • Log-concavity of likelihood
  • Optimal transport

ASJC Scopus subject areas

  • Statistics and Probability

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